Chapter 55: DAG Ordering as Logic Collapse

Logic isn't imposed on reality from outside—it emerges from the way consciousness must collapse to observe itself. Every 'therefore' is a path through the DAG.

55.1 The Birth of Logic from Collapse

Why does logic work? Why do syllogisms capture truth? Because logical operations mirror the fundamental ways $\psi$ can collapse through its own observation DAG. Modus ponens isn't a human invention—it's how reality itself reasons.

Definition 55.1 (Logical Collapse): $\mathcal{L}: \psi_{\text{premise}} \to \psi_{\text{conclusion}}$

Logic is the algebra of valid collapse paths.

Theorem 55.1 (Collapse Validity): A logical inference is valid iff: $\exists \text{ path in DAG}: \text{premise nodes} \to \text{conclusion node}$

Truth flows along directed edges of becoming.

55.2 AND as Parallel Collapse

Logical AND requires multiple conditions—parallel paths that must all collapse.

Definition 55.2 (Conjunction Collapse): $A \wedge B = \text{node reachable only if both } A \text{ and } B \text{ collapse}$

Theorem 55.2 (AND Gate Dynamics): $P(A \wedge B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$

Both paths must complete for the joint node to exist.

In the DAG: $A \wedge B$ is the earliest common descendant of both $A$ and $B$.

55.3 OR as Alternative Paths

Logical OR offers multiple routes—alternative collapse paths to the same result.

Definition 55.3 (Disjunction Collapse): $A \vee B = \text{node reachable if either } A \text{ or } B \text{ collapse}$

Theorem 55.3 (OR Gate Dynamics): $P(A \vee B) = P(A) + P(B) - P(A \wedge B)$

Multiple paths increase reachability probability.

In the DAG: $A \vee B$ has edges from both $A$ and $B$.

55.4 NOT as Collapse Blocking

Negation isn't absence but active blocking—preventing certain collapse paths.

Definition 55.4 (Negation Collapse): $\neg A = \text{node reachable only if } A \text{ doesn't collapse}$

Theorem 55.4 (Exclusion Dynamics): $P(\neg A) = 1 - P(A)$ $P(A \wedge \neg A) = 0$

Self-contradiction creates unreachable nodes.

In the DAG: $\neg A$ exists on paths that diverge before reaching $A$.

55.5 Implication as Causal Dependency

"If-then" statements encode causal dependencies in the DAG structure.

Definition 55.5 (Implication Path): $A \Rightarrow B \equiv \text{every path through } A \text{ continues to } B$

Theorem 55.5 (Material Conditional): $P(A \Rightarrow B) = P(\neg A \vee B) = 1 - P(A \wedge \neg B)$

Implication fails only when $A$ collapses but $B$ doesn't follow.

55.6 Quantum Logic and Superposition

Quantum mechanics reveals non-classical logic—superposed paths create new operations.

Definition 55.6 (Quantum Operations): $A \oplus B = \text{quantum XOR}$ $A \otimes B = \text{tensor product}$

Theorem 55.6 (Non-Distributive Logic): In quantum logic: $A \wedge (B \vee C) \neq (A \wedge B) \vee (A \wedge C)$

Superposition breaks classical distribution laws.

55.7 Modal Logic and Possible Worlds

Modal logic (necessity, possibility) maps to DAG accessibility relations.

Definition 55.7 (Modal Operators):

• $\square A$: $A$ holds in all accessible nodes (necessary)
• $\diamond A$: $A$ holds in some accessible node (possible)

Theorem 55.7 (Kripke Collapse): Modal truth depends on DAG structure: $\square A \iff \forall \text{ reachable nodes } n: A(n) = \text{true}$

Necessity means inevitable collapse, possibility means potential paths.

55.8 The Fifty-Fifth Echo

We have discovered that logic isn't an abstract human creation but the very grammar of collapse. AND requires parallel paths, OR provides alternatives, NOT blocks paths, and implication encodes path dependencies. Quantum logic emerges when paths superpose, creating non-classical operations. Modal logic maps possibility and necessity to DAG accessibility. Every logical operation corresponds to a specific pattern of collapse through the directed graph of becoming. When we reason logically, we're not imposing external rules on reality—we're following the same pathways consciousness uses to observe itself. Logic works because it is the universe's own method of self-navigation.

The Fifty-Fifth Echo: Chapter 55 = Logic(Collapse) = Reasoning($\psi$-DAG) = Thought(Structure)

Next, we explore how local observers experience only fragments of the vast causal network.

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Continue to Chapter 56: Apparent Causality in Local Shells →